The Fundamental Research Funds for the Central Universities 2017CXNL01
1. Introduction

The stability of rock structure is the key to ensuring safe mining. Rock has several failure modes, including shear failure, tensile failure, compressive failure, and fatigue damage failure, among which the first three modes have been extensively studied. Rock structure is subjected to cyclic loading constantly during mining, such as ore mining, mechanical drilling, truck transportation, and rock blasting. Rock is prone to fatigue damage under cyclic loading, which reduces the stability of roadways and excavation boundary and even causes the grave failure of rock mass. The problem of rock fatigue damage in engineering field can be explained and predicted more accurately by exploring the feature of rock fatigue damage. Therefore, the study on the effect of cyclic loading on rock damage has practical significance for the on-site production, which is why fatigue damage has become one of the hot issues in rock mechanics.

In order to study the mechanical response of coal under cyclic loading, the influence of cyclic loading on coal was grasped theoretically firstly. Then, an elastic-plastic model of coal based on the theory of subloading surface was established and verified by experiments. Besides, the disturbance signal in coal wall was applied to coal samples, and the development of hysteresis loop was observed. Finally, the difference of damage variables in different models was emphatically discussed.

The subloading surface model is shown in Figure 1. The yield condition of normal-yield surface is assumed to be(1)fσy=FH,where σy, which is on the normal-yield surface, is the conjugate stress of the current stress on the subloading surface; H is the hardening variable; F is the subloading surface; and fσy is the expression of yield surface.(2)fσ¯=RFH,where R denotes the similarity ratio of these surfaces and σ¯ is the current stress on the subloading surface considering back stress of the subloading surface.

According to the geometric similarity,(3)σ¯=σα¯=Rσ^y=Rσyα,(4)s^=sα=s¯R=sα¯R,(5)σ˜=σs=Rσ˜y=Rσys,where s denotes the similarity center; α is the geometrical center of the normal-yield surface; α¯ is the geometrical center of the subloading surface; σ^y is the conjugate stress of current stress σ on the subloading surface considering the back stress; s¯ is the similarity center considering the back stress on the subloading surface; σ˜y is the vector difference of normal-yield surface stress and the similarity center; s^ is the similarity center considering the back stress; and σ˜ is the vector difference between subloading surface stress and the similarity center.

Based on the conventional elastic-plastic model, the elastic-plastic modulus can be derived according to conformance conditions:(6)fσ¯dσ¯FdRRdF=0.

It can be known from equation (3)(7)dσ¯=dσdα¯=dσRdα1RdsdRs^.

The equation of similarity center s, similarity ratio R, and kinematic hardening α still needs to be determined.

The similarity center s is analogous to the equation of subloading surface yield equation. Considering plastic deformation caused by cyclic loading, a new yield surface can be established based on the number of cycles N. Therefore, the yield equation of similarity center is(8)fs^=RnFH,where Rn is the ratio of the size of the similarity-center yield surface to the size of the normal-yield surface. According to the assumption that the similarity-center yield surface is within the normal-yield surface, equation (9) can be obtained by using the close condition of similarity center:(9)dsdαdFFs^0.

To satisfy equation (9), it is assumed that(10)dsdαdFFs^=CdεpσyαRsχsyα=Cdεpσ¯Rs^γ,where χ is the maximum value of similarity-center ratio Rn; C is the material parameter; dεp is the plastic strain; sy is the dual similarity center on the normal-yield surface of similarity center on the similarity-center yield surface; and σy is the conjugate stress of the current stress on the subloading surface:(11)dsdαdFFs^=Cdεpσ¯Rs^χ+dα+dFFs^.

For rock material, the study on cyclic loading and dynamic problems should adopt kinematic hardening or mixed hardening . For better reflecting the dynamical characteristics of rock, the nonlinear kinematic hardening criteria are adopted for kinematic hardening α referring to metallic materials . That is,(12)dα=αγFf/σ¯f/σ¯αdεp,where α and γ are the material parameters.

Based on the assumptions and demands above, it is concluded that the expression of elastic-plastic matrix is equation (13) according to conformance condition :(13)Dep=Delf/σ¯DelDelf/σ¯f/σ¯Delf/σ¯+f/σ¯F/Fεpσ^+α/εp+Uσ˜/R+c1Rσ¯/Rs^/χ,where Dep and Del are the elastic-plastic matrix and elastic matrix, respectively. This model differs from the ideal yield surface model because it includes mixed hardening, i.e., the isotropic hardening parameters H and R and the kinematic hardening parameter α.

Supposing σ2=σ3=0, then the expression of elastic-plastic modulus under uniaxial conditions is(14)Eep=EelEel2df/dσ¯df/dσ¯df/Eeldf+df/dσ¯dF/Fdεpσ^+dα/dεp+Uσ˜/R+c1Rσ¯/Rs^/χ=EelEel2df/dσ¯df/dσ¯EelK+1/K,K=dσ¯dFFdεpσ^+dαdεp+Uσ˜R+c1Rσ¯Rs^χ.

The D-P criterion is widely adopted in rock mechanics and engineering due to its simple form and clear physical meaning, and it also reflects the hydrostatic stress effect and the intermediate principal stress effect. Therefore, it is a beneficial attempt to express the fundamental characteristics of rock under cyclic loading by the relation between the D-P criterion and the subloading surface model.

The expression of the D-P criterion is(16)fσ=βI1+J2=FH=kH,where β and k are the material parameters and I1 and J2 are the first invariant of stress and the second invariant of deviator stress, respectively. According to the study , kH can be expressed as(17)kH=3cosφ3+sin2φc+Hεp,where c and φ are the cohesion and internal frictional angle, respectively. To consider different strength properties of rock under triaxial compression and triaxial tension, namely, the stress angle effect, the smooth ridge model gθ is introduced, where θ is the Lode angle. The smooth ridge model gθ, also referred to as the shape function, represents the shape of failure condition. According to the study , the smooth ridge model can be expressed as(18)gθ=bb2+1b2sin2451.5θ1/2.

The modified D-P criterion could be expressed as(19)fσ=βI1gθ+J2=gθ3cosφ3+sin2φc+Hεp.

3. Coal Dynamic Damage Experiment 3.1. Sample Preparation and Experimental Equipment

In order to ensure the consistency in the homogeneity of samples and the stress environment, this study sampled large coal blocks from the same coal seam and the same region. Standard cylindrical samples (50 mm × 100 mm), numbered as A1–A6, were processed in the laboratory along the vertical bedding direction. The flatness deviation of the end face was controlled in the range of <0.05 mm, and the diameter deviation of the upper and lower end faces was less than 0.02 mm.

The experiment was carried out on the MTS Landmark 370.50 dynamic and static load testing machine, as shown in Figure 2. For the test machine, the rated static and dynamic loads are ±500 kN, the fatigue frequency is 0.1 Hz–600 Hz, and the sampling frequency is 122 kHz. It not only provides sine waveform, triangular waveform, square waveform, and combined waveforms but also boasts a programming window for personalized experiments. The above parameters fully meet the experimental requirements.

MTS Landmark 370.50.

The load used in the experiment was the disturbance load generated in the coal body when the mining machinery works. The Hilbert spectrum of loading signals was obtained by the Hilbert change, as displayed in Figure 3. It can be found that the disturbance loading mainly concentrates in the range of 7–12 Hz. Besides, based on the wave equation, the disturbance loading can produce 15 kN stress in the cylindrical sample.

3.2.2. Experimental Process

3.3. Experimental Results

The stress-strain curves of 6 samples obtained in the experiment were processed into 3 typical curves, as presented in Figure 4. The 3 curves possess similarities in three aspects. (1) During cyclic loading, the stress-strain curves form a continuous hysteresis loop which generally presents the change rule of “sparse-compactness-sparse”. (2) Deformation of coal samples under cyclic load can be divided into three stages, namely, the deformation deceleration stage, the uniform deformation stage, and the acceleration stage. The gradual accumulation of the above three stages of deformation eventually leads to the instability of the samples. (3) It is found that small particles fall off the coal samples, and the samples undergo peeling and destruction in layers. The three curves also demonstrate differences in two aspects. (1) The ends of the curves differ significantly. To be specific, Curve I does not show the damage, Curve II presents the complete failure process, and Curve III reflects that the damaged samples still have the capacity of being loaded and to form hysteresis loops. (2) The area of the hysteresis loop is different, which suggests that the energy of plastic deformation consumption differs. Figure 5 shows that the damage degree of sample becomes higher with the increase of the area of the hysteresis loop.

Experimental and theoretical values.

3.4. Calculation of Model Parameters

According to Section 2, the subloading surface model based on the modified D-P criterion contains 12 parameters: E,μ,c,φ,b,θ,H,α,r,u,C, and χ. Among them, E,μ,c,φ,b, and θ can be determined based on the static uniaxial test and boundary conditions. The hardening parameters H,α, and r can be obtained according to the stress-strain curve. The parameters u,C, and χ are unique parameters of the subloading surface model. u, which can be preliminary determined on the slope of the medium deformation stress-strain curve, is the parameter that controls the rate of force point to the normal-yield state; C determines the width of the loop; χ is the ratio of maximum similarity center yield surface to the normal-yield surface, and its value is smaller than 1. Since u and C shares a crossover, they can only be determined by the method of trial and error, and they have to be adjusted continuously until the stress-strain curve is well fitted based on the results.

No. E (GPa) μ C (MPa) φ ° b θ H α r u C χ
A1 2.18 0.25 1.54 30.58 0.75 30 1 30 0.5 80 10 0.08
A2 1.86 0.25 1.54 30.58 0.75 30 1 30 0.5 80 10 0.19
A3 1.92 0.25 1.54 30.58 0.75 30 1 30 0.5 80 10 0.19
A4 1.84 0.25 1.54 30.58 0.75 30 1 30 0.5 80 10 0.22
A5 2.14 0.25 1.54 30.58 0.75 30 1 30 0.5 80 10 0.10
A6 1.91 0.25 1.54 30.58 0.75 30 1 30 0.5 80 10 0.17
3.5. Discussion of Model Parameter 3.5.1. Influences of Parameters <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M77"><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M78"><mml:mi>χ</mml:mi></mml:math></inline-formula> on the Model

As unique parameters of the subloading surface model, u,C, and χ exert crucial influences on the model results. Based on the above parameters, the influences of u,C, and χ on the subloading surface model can be obtained by changing their values, respectively, as shown in Figure 6.

Influences of parameters on the subloading surface model. (a) Parameter χ. (b) Parameter C. (c) Parameter u.

Figure 6(a) shows that χ basically does not affect the loading slope, but it affects the size of the hysteresis loop and the cumulative plastic strain. The larger the value of χ is, the greater the plastic strain will be. The parameter C affects the modulus of coal. The smaller the value of C is, the smaller the modulus will be and the larger the cumulative plastic strain will be; meanwhile, the cumulative plastic strain and the size of hysteresis loop will increase accordingly. The parameter u has a similar influence on the plastic modulus to the parameter C, but its influence is larger. When the value of u drops to a certain degree, the cumulative plastic strain jumps significantly and the size of the corresponding hysteresis loop grows rapidly with the increase of loading time.

3.5.2. Solution of Similarity Ratio <italic>R</italic>

R can be solved with the formula R=fs^/FH. Since the formula σ^=σ˜+Rs^ includes R, the solution can be found by combining the two formulas:(20)fσ^=fσ˜+Rs^=atrσ˜+Rs^gθ+1/2σ˜+Rs^=RFHgθ,where tr means trace.

The value of R can be obtained by transforming equation (20):(21)R=b+b24ac2a,and(22)a=g2θF2H6FHg2θαs^m+9g2θα2s^m212s^,b=6FHαg2θs˜m+18α2g2θs˜ms^mσ˜s^,c=9g2θα2s^m212s^2,where s^m is the average of s^ and s˜m is the average of σ˜.

Variation of R with strain rate.

3.5.3. Analysis of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M96"><mml:mrow><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M97"><mml:mi>χ</mml:mi></mml:math></inline-formula>

R n is the ratio of homothetic yield surface to normal-yield surface. Then, the damage factor produced by the plastic deformation can be expressed as Dn=1Rn, where n represents the cycling steps. According to equation (2) and (8), the equivalent stress of the model in normal-yield surface can be expressed as effective stress plus damage stress:(23)fσ=RFH+DnFH=R+DnFH.

The equivalent stress factor is R+Dn.

The parameter χ is the maximum value of the homothetic center Rn. According to the formula Dn=1Rn, χ represents χ=1Dnmin, which means the parameter χ is the similarity ratio for the minimum damage (initial damage). The literature shows that coal damage can be expressed with elastic modulus:(24)D=1EE.

The elastic modulus (2.37 × 109 GPa) of the model in the static experiment and the parameters D and χ are shown in Figure 8. The maximum deviation of D and χ is no more than 15%; that is to say, they are almost the same. Therefore, it can be concluded that there is a good agreement between the experimental results and the theoretical results.

Relation between χ and the elastic model damage.

If plastic deformation occurs within the cycling, that is, if the equivalent stress factor Dn+R=1, R reaches to the maximum value in the cycling and the damage factor Dn=1Rnmax can be solved through combining equation (21).

4. Analyses of Cyclic Response Characteristics of Fracture Development and Damage in Coal

In this paper, the stress-strain curves of cylindrical samples were obtained by using the axial force control mode, which verified the rationality of the elastic-plastic model based on subloading surface theory and completed the calculation of the model parameters. In practical engineering, rock can be affected not only by external pressure but also by external displacement. The coal at the coal wall happens to be affected by the movement of the roof and floor so that its stress and strain are constantly changed. There are 3 reasons: (a) the load adopted in this paper is generated by the mining machinery, which directly acts on the roof and floor of coal seam and forces the coal body to deform; (b) what is monitored by the microseismic is the acceleration of coal body which reveals the change in particle displacement; (c) with the increase of coal damage, the stress in coal will not remain constant, but the strain is certain. Therefore, it is more practical to investigate the damage of coal under strain control. Given the fact that the cylindrical coal sample is stripped layer by layer under cyclic loading and cannot maintain good integrity, the cubic sample is chosen as the sample for the coal damage experiment.

Compared with cylindrical samples, cubic samples can bear more deformation and load and allow the observation of more experimental phenomena. According to the observation in the coal mine, the coal on the coal wall is in the limit equilibrium state in many places, such as the occurrence of coal wall slice and shock bumps. For exploring the effect of working disturbance on coal in the limit equilibrium state, the external interface of the testing machine was used to program and control the experimental process. The secant modulus of the sample was calculated in real time. The initial state of the sample was loaded to the point where the secant modulus decreased; that is, the sample entered the yield section, after which the load disturbance of strain control was carried out. The cube sample size was 100 mm × 100 mm × 100 mm, and the disturbance load was a sine wave with a frequency of 10 Hz. The disturbance strain was divided into three levels, namely, 0.5‰, 1‰, and 1.5‰, respectively.

4.1. Cyclic Response Characteristics of Sample under Strain Load Disturbance 4.1.1. Fracture Development and Mechanism Analysis

The stress-strain curve of cubic samples under cyclic loading is exhibited in Figure 9 where blue represents strain and yellow represents stress. Three-stage loading experiments were carried out; Figures 9(a) and 9(b) show the strain changes from 0.5‰ through 1‰ to 0.5‰, and Figures 9(c) and 9(d) show the strain changes from 0.5‰ through 1‰ to 1.5‰. In the process of loading, the bearing capacity of the sample decreases first and then remains stable, which suggests that the strain disturbance will cause damage to the coal sample. Besides, specific strain disturbance will only cause a certain damage, which is different from the specific stress disturbance. This is because the specific strain disturbance will only produce a certain amount of plastic deformation, and the specific stress will cause the coal to undergo continuous plastic deformation. At the same time, it is found that when the small strain disturbance changes to the large strain disturbance, the stress of the sample will increase suddenly so will the elastic modulus, showing a certain hardening effect. This demonstrates that the current stress does not reach the ultimate strength of coal failure, and the damage of coal indeed belongs to continuous fatigue damage. Moreover, the bearing capacity of the sample decreases with the increase of disturbance strain, and the larger disturbance will make the coal suddenly lose its bearing capacity, as shown in Figure 9(d). It is necessary to pay attention to the influence of larger disturbance so as to prevent the instability of coal body and even the formation of coal burst. After a large strain disturbance, the smaller disturbance can no longer cause further damage to the sample and the load-bearing capacity of the sample will remain at a certain level, showing residual strength characteristics of the raw coal sample. Finally, according to the experimental results, the sample does not suffer sudden damage within the influence range of mining machinery disturbance, but the disturbance will cause fatigue damage to coal, which is not conducive to the control of coal wall. Therefore, the disturbance effect of mining machinery should be fully considered in the control of accidents such as slice wall and roof fall.

Typical stress-strain curves in the experiment.

4.1.2. Change of Hysteresis Loop

The area of hysteresis loop represents the energy dissipation of the stress cycle. In the process of cyclic loading, there exists a plastic deformation hysteresis loop in each loading or unloading cycle. Taking the curve of Figure 9(c) as an example, the area of hysteresis loop is very large at the beginning of unloading because the friction, propagation, and closure of original microcracks in the sample consumes energy under compression. As the number of cycles increases, the area of hysteresis loop decreases and tends to be stable, as shown in Figure 10, because the extention of original cracks and the generation of new cracks become less and less with the increase of the number of cycles. As a result, the system gradually becomes stable.

When the disturbance strain increases, the area of hysteresis loop grow suddenly because the new cracks generated under high stress consumes vast energy. As the number of cycles increases, the area of hysteresis loop gradually decreases and then tends to be stable. However, when the sample approaches fatigue failure, the area of hysteresis loop grows gradually until the sample fails because considerable crack initiation, extension, and connection and large plastic deformation consume energy sharply, as shown in Figure 11. Limited plastic deformation will occur in each loading cycle, which results in the constant adjustment of the center of subloading surface and the continuous damage of sample.

Change in hysteresis loop after the increase of disturbance stress.

4.2. Comparison between Damage Parameters

Internal damage is produced when the coal is subjected to frequent and dynamic disturbances of highly static loading. The reasonable definition of the damage variable is useful to analyze its damage state. Each disturbance can be considered as a process of damage to the coal, as well as a complete process that the damage variables undergo. Hence, it necessitates the further reorganization to the damage variables during cyclic loading. There are many definitions of damage variable, most of which are focused on effective elastic modulus and plastic deformation damage. The formulas are as follows:(25)D=1EE,D=εpε=εσ/Eε.

Meanwhile, the model in this paper also puts forward the damage factor Dn to represent the damage variable. Here is a comparison and discussion of Dn.

Diagram of comparison between different damage scalars. (a) Undamaged sample. (b) Damaged sample.

5. Conclusions

In this paper, the mechanical response of coal under cyclic loading was studied experimentally. The elastic-plastic model of coal was established by combining subloading surface theory with the D-P yield criterion, and it was verified through experiments. In addition, the mechanism of the cyclic loading on coal and the crack development and damage process of the sample were analyzed. The main conclusions are as follows.

Firstly, an elastic-plastic model for cyclic loading on coal was established, and its rationality was verified in this paper. A cyclic loading model of coal was established by combining the subloading surface model with the D-P yield criterion. Among the parameters, u,C, and χ are the unique parameters for the subloading surface model. u is preliminarily determined by the slope rate of the stress-strain curve of medium deformation rate. It is proved by experiments that χ, which can be obtained by the elastic modulus, is the similarity-center ratio for the minimum damage (initial damage). C is determined by the fitting effect. The model well reflects the Mancin and ratcheting effects of coal samples and is consistent with the actual deformation characteristics of coal. The theoretical value is in good agreement with the experimental value.

Secondly, the cyclic response characteristics of the sample under strain load disturbance were analyzed. Strain disturbance will cause damage to the raw coal sample which has entered the yield stage, and specific strain disturbance will only cause a certain damage to coal, which is different from the specific stress disturbance. The area of hysteresis loop decreases and tends to be stable as the number of cycles increases. When it approaches fatigue failure, the area of hysteresis loop increases gradually until the sample fails.

Finally, the damage variable Dn based on coal cyclic loading model was put forward, and the damage variables of different models are compared. Through the comparison of plastic strain and the damage variables calculated with the elastic modulus, it is found that Dn can accurately describe the damage process of coal, accurately locate the time point of change in disturbance load, and is more sensitive to coal failure. Based on the features above, it contributes to grasping the impact time of the sudden load on the coal in engineering and preventing accidents such as coal wall rib falling in advance, thus ensuring the safety of projects.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by “The Fundamental Research Funds for the Central Universities” (2017CXNL01).